Polynomial-growth conjecture for Möbius values of the permutations $\pi_{n,k}$

For $k\le n$, let $\pi_{n,k}$ be the subpermutation of $\kappa_n$ induced by the values $n+1,n+3,\dots,n+2k-1,1,3n+1,2,3n+2,\dots,n,4n,n+2,n+4,\dots,n+2k$. In particular, $\pi_{n,1}=\pi_n$. Polynomial-growth conjecture. For every fixed $k\ge 1$, the absolute value of the Möbius function grows as

$$|\mu(1,\pi_{n,k})|=\Theta(n^{k^2+6k}).$$

This refines the authors' computed example $\pi_n=\pi_{n,1}$ and is supported by their intuition and preliminary results; the source gives no proof or resolution.